The Light FieldPowerPoint PresentationField RadianceSlide 5Properties of Radiance1st Law: Conversation of RadianceSpherical Gantry 4D Light FieldTwo-Plane Light FieldMulti-Camera Array Light FieldThroughput Counts RaysConservation of ThroughputConservation of RadianceQuizRadiance: 2nd LawSlide 16Slide 17Slide 18Parameterizing RaysSlide 20Slide 21Parameterizing Rays: S2 × R2Parameterizing Rays: M2 × S2Incident Surface RadianceExitant Surface RadianceSlide 26Irradiance from the EnvironmentUniform Area SourceProjected Solid AngleUniform Disk SourceSpherical SourceThe SunPolygonal SourceSlide 34Slide 35Slide 38Types of ThroughputProbability of Ray IntersectionAnother FormulationForm FactorRadiosityForm Factors and ThroughputCS348B Lecture 5 Pat Hanrahan, 2005The Light FieldLight field = radiance function on raysConservation of radianceMeasurement equationThroughput and counting raysConservation of throughputArea sources and irradianceForm factors and radiosityFrom London and UptonLight Field = Radiance(Ray)CS348B Lecture 5 Pat Hanrahan, 2005Definition: The field radiance (lumin ance) at a point in space in a given direction is the power per unit solid angle per unit area perpendicular to the directionRadiance is the quantity associated with a rayField RadiancedAdw( , )L x wProperties of RadianceCS348B Lecture 5 Pat Hanrahan, 2005Properties of Radiance1. Fundamental field quantity that characterizes the distribution of light in an environment. Radiance is a function on raysAll other field quantities are derived from it2. Radiance invariant along a ray.5D ray space reduces to 4D3. Response of a sensor proportional to radiance.CS348B Lecture 5 Pat Hanrahan, 20051st Law: Conversation of RadianceThe radiance in the direction of a light ray remains constant as the ray propagates21 1 1 1d L d dAwF =1 21 1 2 22dA dAd dA d dArw w= =1 2L L\ =1dw2dw21d F22d FF = F2 21 2d d1dA2dAr22 2 2 2d L d dAwF =1L2LCS348B Lecture 5 Pat Hanrahan, 2005Spherical Gantry 4D Light FieldCapture all the light leavingan object - like a hologram( , , , )L x y q j( , ) CS348B Lecture 5 Pat Hanrahan, 2005Two-Plane Light Field2D Array of Cameras2D Array of Images( , , , )L u v s tCS348B Lecture 5 Pat Hanrahan, 2005Multi-Camera Array Light FieldCS348B Lecture 5 Pat Hanrahan, 2005Throughput Counts RaysDefine an infinitesimal beam as the set of raysintersecting two infinitesimal surface elementsT measures/count the number of rays in the beam21 221 2dA dAd Tx x=-1 1 1( , )dA u v2 2 2( , )dA u v1 1 2 2( , , , )r u v u vCS348B Lecture 5 Pat Hanrahan, 2005Conservation of ThroughputThroughput conserved during propagationNumber of rays conservedAssuming no attenuation or scatteringn2 (index of refraction) times throughput invariant under the laws of geometric opticsReflection at an interfaceRefraction at an interfaceCauses rays to bend (kink)Continuously varying index of refractionCauses rays to curve; miragesCS348B Lecture 5 Pat Hanrahan, 2005Conservation of RadianceRadiance is the ratio of two quantities:1. Power2. ThroughputSince power and throughput are conserved, Radiance conservedD �DF D F= =D0( )( ) limTT dL rT dTCS348B Lecture 5 Pat Hanrahan, 2005QuizDoes radiance increase under a magnifying glass?No!!CS348B Lecture 5 Pat Hanrahan, 2005Radiance: 2nd LawThe response of a sensor is proportional to theradiance of the surface visible to the sensor.L is what should be computed and displayed.T quantifies the gathering power of the device; the higher the throughput the greater the amount of light gatheredAR Ld dA LTwW= =��AT d dAwW=��ApertureSensorWACS348B Lecture 5 Pat Hanrahan, 2005QuizDoes the brightness that a wall appears to the sensor depend on the distance?No!!Measuring Rays = ThroughputCS348B Lecture 5 Pat Hanrahan, 2005Throughput Counts RaysDefine an infinitesimal beam as the set of raysintersecting two infinitesimal surface elementsMeasure/count the number of rays in the beam21 221 2dA dAd Tx x=-1 1 1( , )dA u v2 2 2( , )dA u v1 1 2 2( , , , )r u v u vCS348B Lecture 5 Pat Hanrahan, 2005Parameterizing RaysParameterize rays wrt to receiver2 2 2( , )dA u vw q f2 2 2( , )d212 2 221 2dAd T dA d dAx xw= =-2 2 2 2( , , , )r u v q fCS348B Lecture 5 Pat Hanrahan, 2005Parameterizing RaysParameterize rays wrt to source1 1 1( , )dA u vw q f1 1 1( , )d221 1 121 2dAd T dA dA dx xw= =-1 1 1 1( , , , )r u v q fCS348B Lecture 5 Pat Hanrahan, 2005Parameterizing RaysTilting the surfaces reparameterizes the rays All these throughputs must be equal.vvgvvg21 21 221 21 12 2cos cosd T dA dAx xd dAd dAq qww=-==r1 1 1( , )dA u vr2 2 2( , )dA u v1 1 2 2( , , , )r u v u vCS348B Lecture 5 Pat Hanrahan, 2005Parameterizing Rays: S2 × R2Parameterize rays byMeasuring the number or rays that hit a shape( , , , )r x y q fw q jq j w q jp===� ��%%2 22( , ) ( , )( , ) ( , )4S RST d dA x yA dA( )A wr%wrSphere:%2 24 4T A Rp p= =Projected areaCS348B Lecture 5 Pat Hanrahan, 2005Parameterizing Rays: M2 × S2Parameterize rays by( , , , )r u v q f2 2( )( , ) cos ( , )M HT dA u v dSq w q jp� �� �=� �� �� �� �� �� �Nr1 4 2 4 31 4 4 42 4 4 43Sphere:2 24T S Rp p= =Crofton’s Theorem:44SA S Ap p= � =% %( , )u v( , )q fNrCS348B Lecture 5 Pat Hanrahan, 2005Definition: The incoming surface radiance (luminance) is the power per unit solid angle per unit projected area arriving at a receiving surface Incident Surface RadiancewwwF�rrg2( , )( , )iid xL xd dAdArdwrCS348B Lecture 5 Pat Hanrahan, 2005Exitant Surface RadianceDefinition: The outgoing surface radiance (luminance) is the power per unit solid angle per unit projected area leaving at surface Alternatively: the intensity per unit projected area leaving a surfacewwwF�rrg2( , )( , )ood xL xd d AdArdwrIrradiance from a Uniform Area SourceCS348B Lecture 5 Pat Hanrahan, 2005Irradiance from the Environment2( ) ( , )cosiHE x L x dw q w=�2( , ) ( , ) cosi id x L x dA dw w q wF =( , ) ( , )cosidE x L x dw w q w=w( , )iL xdAqdwCS348B Lecture 5 Pat Hanrahan, 2005Uniform Area Source2( ) coscosHE x L dL dLq wq wW=== W��%AW%WCS348B Lecture 5 Pat Hanrahan, 2005Projected Solid Anglecos dq wq2cosHdq w p=�dwCS348B Lecture 5 Pat Hanrahan, 2005Uniform Disk Sourcerha%cos 21 0cos21222 2cos coscos22sind drr ha paq f qqpp apW====+��Geometric DerivationAlgebraic Derivation2sinp aW=%sinaCS348B Lecture 5 Pat Hanrahan, 2005Spherical Source222c ossindrRq wp apW===�%rRaGeometric DerivationAlgebraic
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